Null space and column space basis | Vectors and spaces | Linear Algebra | Khan Academy

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  • Опубліковано 23 гру 2024

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  • @moldyluke
    @moldyluke 8 років тому +184

    You explained in 25 minutes what I have been confused about for the past 200 minutes of my class. Amazing

  • @CodSock
    @CodSock 8 років тому +190

    Anybody else have their linear algebra exam coming up too? haha you saved me once again khan academy, very clear and easy to follow.

  • @ThinkPositive00
    @ThinkPositive00 10 років тому +578

    After 100% understood

    • @sam2026
      @sam2026 5 років тому +24

      ThinkPositive00 lol, you have two separate comments. old UA-cam was something else

    • @kozukioden2406
      @kozukioden2406 5 років тому +35

      I love how both comments have the same exact number of likes ! Math students are so precise lmao

    • @hugoirwanto9905
      @hugoirwanto9905 4 роки тому +1

      @@kozukioden2406 wow 5 months later and its still have the same number of likes

    • @shawnjames3242
      @shawnjames3242 4 роки тому +1

      @@hugoirwanto9905 It still has the same number of likes 223. How far will it go? I am curious...

    • @vishakamohan5336
      @vishakamohan5336 4 роки тому

      @@shawnjames3242 Yes. It's 259 on both now

  • @ThinkPositive00
    @ThinkPositive00 10 років тому +578

    Before 0% understood

  • @ChristinaWahlquist-h5z
    @ChristinaWahlquist-h5z Місяць тому +1

    Thank you! -Christina, 48 year old mother of 7 children finishing a Secondary Math Ed I started 30 years ago. :) Your solution was clear and organized. Awesome!

  • @dripminic
    @dripminic 5 років тому +141

    A faster way to find the basis for the column space is to rref and then take the column vectors with pivots

  • @みかちゃん-k4r
    @みかちゃん-k4r 3 роки тому +8

    I'm just gonna say again, I don't really understand what my professor said but I'm able to understand the explanation from this video. It really helped me a lot, no matter I'm gonna fail this subject or not, thank you for making this video.

  • @ozzyfromspace
    @ozzyfromspace 5 років тому +13

    The number of pivot variables = number of independent basis vectors that make up the column space of A. Very insightful, Sal! It took me a while to process but now I get it ☺️

  • @benjaminjongepier6826
    @benjaminjongepier6826 10 років тому +27

    I love you Khan Academy

  • @certifiedwavy
    @certifiedwavy 10 місяців тому +2

    thanks, i do not why i could not understand this but your video did the trick!

  • @Lolsashalol
    @Lolsashalol 5 років тому +17

    i've got a feeling that i'll get my bachelors in Mech Engineering with this channel

  • @EwaldBE
    @EwaldBE 4 місяці тому

    I have always heard good things about Khan Academy and it definitely checks out. This video explained a topic I have been struggling with clear as day.

  • @ryansuber6253
    @ryansuber6253 3 місяці тому

    I understood more in this 25-minute video than in my lecture on the topic today........ Thank you for this video

  • @artindesign2565
    @artindesign2565 3 роки тому +1

    Ohhhhhhhh thankxxxx a lot....!! Finally I understand the difference of null and column space and it works for creating basis.

  • @liliashaymuratova6729
    @liliashaymuratova6729 Місяць тому

    It's fantastic! So straight-forward. Thank you so much!

  • @unnamed1992
    @unnamed1992 13 років тому +2

    OMG YOUR A GENIUS. I CAN'T BELIEVE I LEARNED THAT.

  • @4sky
    @4sky 13 років тому +10

    2am in morning..."ill let you go for now"
    "yes!! im free! i can go to sleep!"

  • @supersonic174
    @supersonic174 6 років тому +9

    so if there are free variables in the reduced row echelon form, does that mean that it is linearly dependent

  • @GbyP
    @GbyP 10 місяців тому

    This man has saved so many people's grades, about to take my linear algebra midterm rn 😅

  • @oliverandamms738
    @oliverandamms738 2 місяці тому

    thank you !! i know im not alone when i say that! this cleared everything up so neatly

  • @theekags
    @theekags 7 місяців тому +1

    thank you so much i have a final in 4 hours and this made everything simpler

  • @ccuuttww
    @ccuuttww 6 років тому +2

    the last part may not necessary to find the basis u can just pick it form the reduced encholen form which have pivot in each column in this case it is column 1 and 2

    • @zoomboy6676
      @zoomboy6676 5 років тому

      But he just proved that columns 1 and 2 are sufficient for finding the basis

  • @tejasghodkhande3381
    @tejasghodkhande3381 4 роки тому +1

    Very Nice explanation!

  • @reypope19
    @reypope19 13 років тому +2

    You're saving my linear algebra grade, THANKS!

    • @Novice0825
      @Novice0825 4 роки тому +2

      I assume you've graduated by now!

  • @cjames9001
    @cjames9001 14 років тому +3

    this 25 minute lecture puts 3 weeks of lecture in class to shame, very helpful

  • @RawwestHide
    @RawwestHide 9 років тому +8

    khan is a god

  • @hrzbltnrd
    @hrzbltnrd 9 місяців тому

    thank you so much, finally a video i can understand

  • @NotmyYTchannel
    @NotmyYTchannel 15 років тому +1

    OMG... I was just on this studying this topic right now... and you posted this up like 10 minutes ago... WOW!!

  • @vishnus2567
    @vishnus2567 4 роки тому +3

    When we do the echelon reduction, do we need to make sure that the pivot elements need to be 1?

  • @khanacademy
    @khanacademy  15 років тому +12

    That's not exactly giving me the best incentive to finish

    • @deryakarabulut7805
      @deryakarabulut7805 5 років тому +1

      Hello, is there not a mistake done in the first place when you were subtracting 2 times row 1 from row 2? You said so but you subtracted row 2 from 2 times row 1 and it changed all the result. I try to understand linear algebra and everything coming up with it so I may be wrong but this is opposite to what I learned from MIT open courseware and what you said in this very video. Please clarify this point for me or I ill get lost!

  • @rajaabubakar4104
    @rajaabubakar4104 7 років тому

    this video should be of maximum 5 mins....but u are awesome in extending videos

  • @devashishbhake3173
    @devashishbhake3173 3 роки тому

    this video is pretttttyyyyyyy old yet very relevant in 2021......

  • @MrCalhoun556
    @MrCalhoun556 14 років тому +1

    I think it makes a bit more sense to apply Elementary Row Operations upon the Matrix before figuring out the Column Space. You'll see already before if the system of equations collapses the vector to a line, plane or 3d hyper-plane. It also has then a nicer form to check for the results of the Rank-nullity theorem.

    • @ArthurTaylor
      @ArthurTaylor Рік тому

      So when the determinent is zero, the system of equations collapses down to a line?

  • @梁廷睿-t5k
    @梁廷睿-t5k 3 роки тому

    Great!

  • @shameerrishad4189
    @shameerrishad4189 2 роки тому +1

    I have a query: are pivot variables aka dependent variables & free variables aka independent variables?

  • @patrickneal9288
    @patrickneal9288 2 роки тому

    this saved my life

  • @JeremyLeeTW
    @JeremyLeeTW 7 років тому

    great for the review of basis, null space and column space for a matrix !

  • @jacobm7026
    @jacobm7026 6 років тому

    Mind. Blown.

  • @tugbamacit4224
    @tugbamacit4224 6 років тому

    adamsın adam!! (trying to get it for a day long. finally you made it. thanks in advance.)

  • @drrojas
    @drrojas 13 років тому

    KHAN ACADEMY in HD , aaawwww yea!!

  • @Asdun77
    @Asdun77 4 роки тому

    You explained it very easy thank you, god bless you

  • @TDefton
    @TDefton 5 років тому

    So in order for the column space to be Liniarly independent, the rref would have to be the identity matrix, right?

  • @arjunselvam7
    @arjunselvam7 Рік тому

    This is the single most redundant way to explain that pivot variables determine the column space but I finally got it

  • @shriram6123
    @shriram6123 2 роки тому

    Very nicely explained

  • @Liaomiao
    @Liaomiao 12 років тому +1

    are pivot variables always the linearly independent ones? can't you write the pivot variables in terms of the free variables here as well? ack it's kinda coming together for me... thx khan

    • @aryankumarprasad1574
      @aryankumarprasad1574 3 роки тому +1

      are pivot variables always the linearly independent ones- Yes

  • @vincelunceford
    @vincelunceford 11 років тому

    yeah i totally agree... but he tries to prove it more theoretically

  • @hansgodoy6434
    @hansgodoy6434 5 років тому

    thank u very much

  • @SaeedRanjbar
    @SaeedRanjbar 10 років тому +1

    anaaaaaaaaaaaaaaaaaaaazing video ! Neat Clear , thanks !

  • @콘충이
    @콘충이 4 роки тому

    Thanks!

  • @linkmaster959
    @linkmaster959 5 років тому +1

    Can the basis of the column span be the columns with pivots in rref?

  • @GaryTugan
    @GaryTugan 3 роки тому

    awesome vid

  • @oneinabillion654
    @oneinabillion654 5 років тому

    Took me 1 day to understand span subspace basis null space column space and then remembering it

  • @iczyg
    @iczyg 11 років тому

    Can a vector be in both a the Null space AND the Column space of some set of vectors? Or is it one or the other...?

  • @kenikozo
    @kenikozo 13 років тому

    ITS MAGIC!!!

  • @meghnadsaha2469
    @meghnadsaha2469 10 років тому +1

    YEA IT IS MOST IMPORTANT FPR EVERYONED , BY THIS WAY I THIK ANYBODY CAN LEARN MATH S BIN SIMPLE WAY

  • @rkishei
    @rkishei 12 років тому

    I wouldn't say it's so much over-explanation rather than thinking out loud. At least for me, this helps, not because I don't know how to subtract (subtraction being one of many things he 'over-explains'), but because I can keep track of every assumption he's making.

  • @PrinceFX
    @PrinceFX 15 років тому

    AWESOMENESS !!!

  • @akshitajohar16
    @akshitajohar16 2 роки тому

    Where are next videos , please tell can't find them

  • @Europemaster
    @Europemaster 14 років тому +1

    @khanacademy
    he is probably being sarcastic or just a throll, you are doing amazing job with your amazing explanations, dont let that anonymous idiots make you lose strength to carry on. Have a nice day.

  • @hasunsri
    @hasunsri 11 років тому

    most probalably....self study...........or.........one good teacher(lecture) who knows the subject deeply....not by just passing the exams.....by feeling maths....

  • @alepov
    @alepov 12 років тому

    thanks

  • @SouthernHadoken
    @SouthernHadoken 6 років тому +1

    there easier way to figure out the basis. it is the original columns that correspond to the pivot columns in its RREF.

  • @roelheirman8398
    @roelheirman8398 9 років тому +12

    You just saved my ass :)

  • @vatcherc
    @vatcherc 13 років тому

    THANK YOU!!!!

  • @manpreetsaggi786
    @manpreetsaggi786 11 років тому

    It's not you, it's just the human nature that can't accept the truth and the truth is majority of the teachers here don't care if the student learns or not.(not all cuz I have some great Profs at my school). But most teachers here just work for their pay check. That doesn't happen in India. People care more about each other.
    Now this guy explaining everything for free, that's the kind of spirit we need in teachers her. I don't want them to teach for free but just care more than they do..

  • @ThePearReviews
    @ThePearReviews 11 років тому

    Its easier to say that the pivot columns of A form a basis for Col(A) :P

  • @dezebarrow3663
    @dezebarrow3663 3 роки тому +1

    Even though i finished this video, i play it back just to hear his voice :'(

  • @lemyul
    @lemyul 5 років тому

    thanks sal sal

  • @VicfredSharikver
    @VicfredSharikver 15 років тому

    nice

  • @iLoveTurtlesHaha
    @iLoveTurtlesHaha 7 років тому

    1:34 "We don't know that these are linearly independent" ... yes we do, there are 4 vectors in R3, one of them will be redundant, therefore those vectors are linearly dependent. Also, after you row reduced, you just needed to see which columns had a pivot point then go back to the original matrix and take those columns and they are the basis vectors for Col(A). ... eg. there was a pivot position in the columns for x1 and x2, the basis vectors were eventually determined to be column 1 and column 2 of the original matrix. Make sure you understand what is going on in the video though, it's really important that you do.

  • @martinmarmo
    @martinmarmo 8 років тому

    Very enlightening video! One question though. What software do you write on? I'd love to take notes in class using the same method

  • @javierzanet
    @javierzanet 14 років тому

    Well because you have 4 vectors in R3 so you can tell that they are linearly dependent.

  • @aaad1100
    @aaad1100 8 років тому

    Curious, when you first proved that X3 & X4 were "free" variables, is that enough evidence to consider those vectors redundant and exclude them from the final linear independent set, or was that just coincidence?

    • @faisaladel5034
      @faisaladel5034 8 років тому

      +aaad1100
      It's even more than that ,seeing that in the reduced echelon form that the non zero rows are just 2 ,and the number of columns (variables) is 4 ,then you should figure out there is two additional variables or additional
      redundant vectors.

  • @andreashaugstvedt8076
    @andreashaugstvedt8076 5 років тому

    What happens if you have a column consisting of only 0's, regarding the null space basis? Wouldn't that mean that the respective x-variable is neglectable?

  • @orpheuspericles9582
    @orpheuspericles9582 7 років тому +1

    shouldn't the no. of basis vectors be equal to the dimension of the subspace??

    • @vishalgoel6690
      @vishalgoel6690 7 років тому

      Orpheus Pericles No, because here you can see that he put 0 for x3 while proving that v4 is redundant and put 0 for x4 while proving that v3 is redundant. So, we can get rid of both v3 and v4. Also, the basis of a subspace need not span all the points in the graph because the span of the subspace can be limited. For example, here, the span is limited to a plane in R^3. What we can say is that the number of vectors in basis need not be greater than the order of dimension.

    • @ahmeddesoky8434
      @ahmeddesoky8434 7 років тому

      For the point you mentioned @Vishal Goel, " the basis of a subspace need not span all the points in the graph ".... I think it is not as per the definition Sal gave in a previous video that the basis is the minimum set of vectors that spans the subspace !
      Also, till now I am not totally convinced how the number basis vectors of a subspace to be less than the subspace order !?

    • @ahmeddesoky8434
      @ahmeddesoky8434 7 років тому

      The next video explains and visualizes that point. Thanks !

  • @DrinkedTooMuch
    @DrinkedTooMuch 2 роки тому

    So we have weird exercises to do as homework (tho we havent even done ANY exercises on this topic, all they did was throw empty definitions at us and expect us to be geniuses) where it says
    "Which vectors(b1,b2,b3) are in the column space of A?"
    A= 1 1 1
    1 2 4
    2 4 8
    And thats all the info we have. How does one solve it?

  • @unfragger
    @unfragger 15 років тому

    I LLOVE YOU

  • @TBV121
    @TBV121 13 років тому +1

    I think you made a mistake on your second computation. -2 x Row 1 added to the remainder of the entries in Row 2 should give -1, 2 and 1, not 1, -2 and -1.

  • @rituparnameshram9397
    @rituparnameshram9397 3 роки тому

    who are those ultra genius 93 people who disliked this video?

  • @manpreetsaggi786
    @manpreetsaggi786 11 років тому

    Dear friend he is talking about the education standards of the US which are very very low as compared to other countries. What you are given in 12 grade her, I was given that stuff in 9th in India

  • @MohamedElsheikh22
    @MohamedElsheikh22 12 років тому

    The basis of Nul(A) is the same spanning set of Nul (A)...
    I think you forget to say that!

  • @human.earthling
    @human.earthling 13 років тому

    haha, at 0:06 ...CURL over... ..really INTEGRATE everything...

  • @sakhatbanda1529
    @sakhatbanda1529 7 років тому +5

    if you speed this up to 1.5 , it essentially feels like a man trying to win an argument against a whamen

  • @imbsalstha
    @imbsalstha 12 років тому

    thanks again ! well ,i'm gonna forget mine LA teacher but not you.

  • @yuanguolang5352
    @yuanguolang5352 8 років тому

    any one could help me to find the basis of left nullspace?

  • @manpreetsaggi786
    @manpreetsaggi786 11 років тому

    An average kid here need a calculator, an equation sheet for an exam and it's provide, where as any of that stuff in Indian schools is strictly prohibited. I am not talking about the small schools in the poor villages. I am talking about the prestigious schools which we have many

  • @cvpadre
    @cvpadre 11 років тому

    Thanks for the video. Hope you keep up the good work, which obviously you are =0)

  • @ArthurTaylor
    @ArthurTaylor Рік тому +1

    How did I pass this subject? This is so confusing 😭

  • @Warrimonk
    @Warrimonk 14 років тому

    Very helpful thanks, too bad I find it impossible to stay away in any sort of linear algebra lesson *yawn*

  • @slottedaloha649
    @slottedaloha649 5 років тому +1

    About getting RREF, you've made a mistake (that actually not criticall, but anyway), when you subtracted 2 times row 1 from row 2 you said the one thing and did another one, you didn't subtract 2xR1 from R2 but added 2xR1 to -R2

  • @pianoforte17xx48
    @pianoforte17xx48 4 роки тому

    *nullsapce*

  • @bojanglessr3
    @bojanglessr3 11 років тому

    to moeb32, he said he was doing 2r1-r2 not r2-2r1...

  • @louaialfaori7978
    @louaialfaori7978 11 років тому

    a Gizzillion Times agreed!!!!!

  • @joodmu2002
    @joodmu2002 Рік тому

    I love u

  • @realvideosrv1879
    @realvideosrv1879 4 роки тому

    At the end, didn't he mean to say column space of A "C(A)" ? Instead of column span of A?

  • @bakhtiareng.6392
    @bakhtiareng.6392 5 років тому

    SALL KHAN is proud of MUSLIMS

  • @Clodidi1
    @Clodidi1 3 роки тому +2

    This doesn't actually teach you what a null space is.. this basically teaches you some trick to figure out the basis of a subspace.... waste of 20 mins.

    • @majed1911
      @majed1911 2 місяці тому

      There is a separate video for that

  • @teomazzaferro7040
    @teomazzaferro7040 12 років тому

    just because people are in linear algebra doesnt mean they can follow simple calculations, there's some people in my class that are really dumb

  • @jojogaroot
    @jojogaroot 8 років тому +9

    there's a mistake when you row reduced the matrix

    • @StirsMYCookiez
      @StirsMYCookiez 8 років тому

      +Abdulmajeed Garoot ?

    • @ericroncin436
      @ericroncin436 8 років тому +1

      If you're talking about the result of row 2 in the first step, he did the calculations and then multiplied the row by -1 to make his leading one positive. He just never said it.

    • @AgueroIsKing
      @AgueroIsKing 8 років тому +1

      No he didn't. You can multiply rows and columns by scalars, it doesn't change anything.

    • @abdullahkardas8887
      @abdullahkardas8887 6 років тому

      thanks for your comment

  • @Sythesia
    @Sythesia 6 років тому

    Null Sapce